Milk intolerance in babies
Both bottle and breast fed babies can react to milk. The symptoms may be lots of crying, projectile vomiting, excessive spitting up shortly after meals, very sore diaper rashes that won't clear up and rigid body while crying which may be an indicator of stomach pain. If you are concerned please mention the symptoms to the baby's pediatrician.
Dairy protein intolerance and milk/soy protein intolerance both manifest in the ways Melissa described above. They're painful and can be mistaken for GERD/reflux. Or they can even be causing GERD/reflux. Your baby can even react to those proteins in breastmilk from the food that you eat. It can be difficult to adjust your diet and harder still not to slip up.
A true dairy allergy is a bit different. The symptoms Melissa mentioned will be involved, but you will likely also see mucus-y diapers, blood in the diaper, eczema or other types of skin rash, failure to maintain growth curve, and even excessive fluid in the ears that can contribute to ear infections and (usually temporary) hearing difficulties. Anaphylaxis, the most severe type of reaction, includes symptoms such as shortness of breath, coughing, wheezing, swelling, diarrhea, hives, dizziness, and sudden panic. These symptoms usually set in quickly and progress fast. If your child experiences this, call an ambulance immediately.
Lactose intolerance is extremely rare in babies. It's pretty normal in adults though. Most babies who are intolerant to dairy are intolerant to the proteins (casein), not the sugars (lactose). I'm not saying it doesn't happen, I am just saying it is rare and often misdiagnosed. Not that it really makes a difference as long as you avoid dairy either way.
There are also a few diseases and syndromes that can manifest as a bad reaction to dairy. They're pretty rare overall though.